The direction ratios of the normal to the pla | vedclass

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(D) The equation of the family of planes passing through the line of intersection of the planes $x+2y+3z-4=0$ and $4x+3y+2z-1=0$ is given by $(x+2y+3z

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$3, 2, 1$

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(D) The equation of the family of planes passing through the line of intersection of the planes $x+2y+3z-4=0$ and $4x+3y+2z-1=0$ is given by $(x+2y+3z-4) + \lambda(4x+3y+2z-1) = 0$. Since the plane passes through the origin $(0, 0, 0)$,we substitute $x=0, y=0, z=0$ into the equation: $(0+0+0-4) + \lambda(0+0+0-1) = 0 \Rightarrow -4 - \lambda = 0 \Rightarrow \lambda = -4$. Substituting $\lambda = -4$ back into the family equation: $(x+2y+3z-4) - 4(4x+3y+2z-1) = 0 \Rightarrow x+2y+3z-4 - 16x - 12y - 8z + 4 = 0 \Rightarrow -15x - 10y - 5z = 0 \Rightarrow 3x + 2y + z = 0$. The direction ratios of the normal to the plane $ax+by+cz+d=0$ are $(a, b, c)$. Thus,the direction ratios of the normal to the plane $3x+2y+z=0$ are $3, 2, 1$.

The direction ratios of the normal to the plane passing through the origin and the line of intersection of the planes $x+2y+3z=4$ and $4x+3y+2z=1$ are . . . . . .

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